On Computing and Pricing of Adjustable Robust Chemical Process Designs

TL;DR

This work focuses on identifying designs that hedge against uncertainties in model parameters to ensure feasibility, taking the possibility to adjust operating conditions into account, and proposes a method to quantify the cost or price of robustness.

Abstract

Model-based process simulation can be used to derive designs and operating conditions of chemical processes that optimally balance multiple objectives, such as quality, costs, or environmental impacts. This work focuses on identifying designs that hedge against uncertainties in model parameters to ensure feasibility, taking the possibility to adjust operating conditions into account. An adaptive scheme is proposed to pinpoint the relevant scenarios in a discretized uncertainty space; these scenarios are then fed into a multi-objective adjustable robust optimization framework reducing the computational burden compared to the consideration of all potential scenarios. Furthermore, we propose a method to quantify the cost or price of robustness, i.e., the compromise which has to be made in comparison to the nominal design case in order to hedge against uncertainty. The conceptual findings are illustrated with an industrially relevant case study.

Figures (12)

• Figure 1: Reference discretizations for (a) a box-shaped and (b) an elliptical uncertainty set $U$: box vertices and face mids for the box-shaped uncertainty set and piercing points in coordinate directions as well as in direction of the space diagonals for the elliptical uncertainty set. The nominal scenario is marked in black and can, but does not have to lie in the center of the uncertainty set.
• Figure 2: Sketch of adaptive scenario choice for three Pareto points a, b, c. The scenarios are chosen from a reference discretization (black, gray, and white points, where the black point represents the nominal scenario) according to Alg. 1. For Pareto point c the union of the (blue) worst-case scenarios for point a and point b is chosen as initial set of worst-case scenarios.
• Figure 3: Illustration of the price of robustness in the MARO setting. The image of a MARO optimized HNV $x^*$ is a point $F^{MARO}(x^*)$ on the MARO Pareto front. Re-optimization of WSV $y$ for the nominal scenario $u_{\text{nom}}$ leads to an improved point $F^{NSR}(x^*)$. The direction of improvement can be extended to a point $F^{MO}$ on the nominal front. The price of robustness $p^R$ is the distance from $F^{MO}$ to $F^{NSR}(x^*)$.
• Figure 4: Illustration of the price of robustness displayed on one of the navigation sliders. The worst-case objective value for the MARO solution is represented by the green hourglass selector, and can be improved by dragging the selector to the left. Yellow and purple markers indicate the objective value for the re-optimized solution (NSR) and the non-robust solution (MO), respectively. Their difference is the price of robustness for this particular objective.
• Figure 5: Separation of a Methanol - Methyl formate mixture using a distillation column.